Linear Equations of Higher Order

The chapter starts with a short summary, without proofs, of some basic facts from linear algebra that will be used. Linear dependence and independence of arbitrary functions, spanning of a linear function space, basis, and dimension are defined. For an nth-order homogeneous linear differential equation which satisfies the existence and uniqueness theorem, we characterize when a set of solutions spans the solution space and when are they independent and show how to obtain such sets. The idea of Wronskian and its properties appear naturally. Wronskian and Abel’s formula are used to construct a differential equation whose solutions are given, show the Sturm separation theorem, and more. Reduction of order is developed for second-order and nth-order equations. The solutions of equations with constant coefficients are developed when the roots of the characteristic equation are real or complex or different or some of them coincide. The same is done for Euler’s equations. Nonhomogeneous equations of nth order are solved by variation of parameters. For second-order equations the Cauchy kernel is introduced, as well. For problems in which the homogeneous equation has constant coefficients and the right-hand side consists only of polynomials, exponentials, sines and cosine, we introduce and verify the method of undetermined coefficients.